% \par \textbf{书上习题：习题6.1 第3题；习题6.2 第1题（2）（4）,第4题}
\chapter{第二周作业}
\begin{remark}\label{normal_calculus}

常用不定积分：
\vspace{1em}
    \begin{enumerate}
       
\item $\begin{aligned}
    \int\frac{\mathrm{d}x}{x^2+a^2}=\frac{1}{a}\arctan\frac{x}{a}+C(a>0).\end{aligned}$ \vspace{1em}
\item $\begin{aligned}\int\frac{\mathrm{d}x}{x^2-a^2}=\frac1{2a}\ln\left|\frac{x-a}{x+a}\right|+C(a>0).\end{aligned}$\vspace{1em}
\item  $\begin{aligned}\int\frac{\mathrm{d}x}{\sqrt{a^2-x^2}}=\arcsin\frac xa+C(a>0).\end{aligned}$\vspace{1em}
\item $\begin{aligned}\int\frac{\mathrm{d}x}{\sqrt{x^{2}\pm a^{2}}}=\ln|x+\sqrt{x^{2}\pm a^{2}}|+C(a>0).\end{aligned}$
\vspace{1em}
\item $\begin{aligned}\int\ln x\mathrm{d}x=x\ln x-x+C.\end{aligned} $
\vspace{1em}
\item $\begin{aligned}\int\sec x\mathrm{d}x=\ln|\sec x+\tan x|+C \end{aligned}$
\vspace{1em}
\item $\begin{aligned}\int\csc x\mathrm{d}x=-\ln|\csc x+\cot x|+C.\end{aligned} $
\vspace{1em}
\item $\begin{aligned}\int\sqrt{x^{2}\pm a^{2}}\mathrm{d}x=\frac{1}{2}\left[x\sqrt{x^{2}\pm a^{2}}\pm a^{2}\ln|x+\sqrt{x^{2}\pm a^{2}}|\right]+C(a>0); \end{aligned}$
\vspace{1em}
\item $\begin{aligned}\int\sqrt{a^{2}-x^{2}}\mathrm{d}x=\frac{1}{2}\left[x\sqrt{a^{2}-x^{2}}+a^{2}\arcsin\frac{x}{a}\right]+C(a>0)\end{aligned} $
\vspace{1em}
\item $\begin{aligned}\int e^{ax}\cos bx\mathrm{d}x=\frac{e^{ax}}{a^{2}+b^{2}}(a\cos bx+b\sin bx)+C(ab\neq0); \end{aligned}$
\vspace{1em}
\item $\begin{aligned}\int e^{ax}\sin bx\mathrm{d}x=\frac{e^{ax}}{a^2+b^2}(a\sin bx-b\cos bx)+C(ab\neq0). \end{aligned}$
\vspace{1em}
\item $\begin{aligned}\int x\cos nx\mathrm{d}x={\frac{1}{n^{2}}}\cos nx+{\frac{x}{n}}\sin nx+C(n\neq0);  \end{aligned}$
\vspace{1em}
\item $\begin{aligned}\int x\sin nx\mathrm{d}x=\frac{1}{n^{2}}\sin nx-\frac{x}{n}\cos nx+C(n\neq0).\end{aligned}$
\end{enumerate}
\end{remark}
\section{例题(明天讲，不是作业)}
\begin{example}\label{ex:2.1}
    设$f$为连续函数.求证:
    \begin{enumerate}
        \item $\begin{aligned}
            \int_0^{\pi/2}f(\cos x)\mathrm{d}x=\int_0^{\pi/2}f(\sin x)\mathrm{d}x;
        \end{aligned}$
        \vspace{1em}
        \item $\begin{aligned}
            \int_0^\pi xf(\sin x)\mathrm{d}x.=\frac\pi2\int_0^\pi f(\sin x)\mathrm{d}x.
        \end{aligned}$
    \end{enumerate}
\end{example}
\begin{example}
$\begin{aligned}
    \int_0^1x^n\ln x\mathrm{d}x\left(n\in N^+\right)
\end{aligned}$
\end{example}
% \begin{example}
% 求下列积分：
%     \begin{enumerate}
%         \item $\begin{aligned}
%         \int_0^1 \frac{1}{x^3+1}\mathrm{d}x
%         \end{aligned}$
%     \vspace{1em}
%         \item $\begin{aligned}
%         \int_0^1 \frac{1}{x^4+1}\mathrm{d}x
%         \end{aligned}$
%         \vspace{1em}
%         \item $\begin{aligned}
%         \int_0^1 \frac{1}{x^6+1}\mathrm{d}x
%         \end{aligned}$
%     \end{enumerate}
% \end{example}
\begin{example}
    $\begin{aligned}\int_0^{\pi/2}\frac{\cos x\sin x}{a^2\sin^2x+b^2\cos^2x}\mathrm{d}x\mathrm{~(}ab\neq0).\end{aligned}$
\end{example}

\begin{example}
    计算下列积分：
    \begin{enumerate}
        \item $\begin{aligned}
            \int_0^{\pi/2}\frac{\cos^2x}{\cos x+\sin x}\mathrm{d}x;
        \end{aligned}$
        \vspace{1em}
        \item $\begin{aligned}
            \int_0^{\pi/2}\frac{\sin^2x}{\cos x+\sin x}\mathrm{d}x.
        \end{aligned}$
    \end{enumerate}
\end{example}
\section{计算题}
\begin{enumerate}[(1)]
    \item $\begin{aligned}
        \lim \limits_{n \rightarrow \infty}\left(\frac{n}{n^{2}+1^{2}}+\frac{n}{n^{2}+2^{2}}+\cdots+\frac{n}{n^{2}+n^{2}}\right)
    \end{aligned};$
    \vspace{1em}
    \item $\begin{aligned}
        \lim\limits _{n \rightarrow \infty} \frac{1^{p}+2^{p}+\cdots+n^{p}}{n^{p+1}}(p>0)
    \end{aligned} .$
        
    \vspace{1em}
    \item $\begin{aligned}
        \int_{1}^{4} \frac{x+1}{\sqrt{x}} \mathrm{~d} x 
    \end{aligned};$
    \vspace{1em}
    \item $\begin{aligned}
        \int_0^{\sqrt{3}}x\arctan x\mathrm{d}x
    \end{aligned};$
    \vspace{1em}
    \item $\begin{aligned}
        \int_{-1}^0(2x+1)\sqrt{1-x-x^2}\mathrm{d}x
    \end{aligned};$
    \vspace{1em}
    \item $\begin{aligned}\int_0^a\ln{(x+\sqrt{a^2+x^2})}\mathrm{d}x\mathrm{~}(a>0)
    \end{aligned};$
    \vspace{1em}
    \item $\begin{aligned}
        \int_0^{\ln2}\sqrt{\mathbf{e}^x-1}\mathrm{d}x\text{ ;}
    \end{aligned}$
    \vspace{1em}
    \item $\begin{aligned}    \int_0^{\pi/4}\frac{\mathrm{d}x}{\cos x}\end{aligned}.$
    \vspace{1em}
    
\end{enumerate}
\section{证明题}
\begin{enumerate}[1.]
    \item 设$I$是一个开区间，函数$f$在$I$上连续，并且$a<b,a,b\in I.$求证：
$$
\lim_{h\to0}\frac1h{\int_a^b(f(x+h)-f(x))\mathrm{d}x}=f(b)-f(a).
$$
    \item (Fejér积分) 求证：$\begin{aligned}
       \int_0^\pi\left(\sin\frac{nx}2/\sin\frac x2\right)^2\mathrm{d}x=n\pi\quad(n=0,1,2,\cdots).
    \end{aligned}$
    
    提示：等价地求$\begin{aligned}
        \int_0^{\frac\pi2}\frac{(\sin nx)^2}{\sin^2x}\mathrm{d}x
    \end{aligned}$,
    
    由$\begin{aligned}
        \sin^2x-\sin^2y=\sin(x-y){\sin(x+y)}
    \end{aligned}$，可知
    
    $\begin{aligned}
        \int_0^{\frac\pi2}\frac{(\sin nx)^2}{\sin^2x}\mathrm{d}x=\int_0^{\frac\pi2}\frac{[\sin(n-1)x]^2+\sin(2n+1)\mathrm{sin}x}{\sin^2x}\mathrm{d}x=\int_0^{\frac\pi2}\frac{[\sin(n-1)x]^2}{\sin^2x}\mathrm{d}x+\frac\pi2
    \end{aligned}$
    
    上式第二个等号利用了$\begin{aligned}
        \int_0^\pi\frac{\sin\left(n+\frac12\right)x}{\sin\frac x2}\mathrm{d}x\mathrm{~}=\pi\quad(n=0,1,2,\cdots)
    \end{aligned}$，要证此积分，需要考虑$\begin{aligned}
        \sum_{k=1}^n \cos(kx)\text{的结果}
    \end{aligned}$
\end{enumerate}
